Numerical Methods

For a long time I’ve held negative views towards numerical methods, particularly “trial and improvement” and the trapezium rule, but I’ve been reconsidering those views. This has been quite a long process that began when Tom Bennison (@DrBennison) questioned negativity towards them, probably around a year ago. We had a brief discussion around them and some of the thoughts have been stewing since.

Tom reminded me that numerical methods are important as in the real world there are many things that cannot be done another way (yet!). The discussion left me thinking that rather than numerical methods themselves being bad, it’s could be more to do with the way they are framed.

I remember when I was studying towards my own A levels I was taught the trapezium rule for numerical integration. My teacher said it was what was used before calculus was invented and that it had no real use now but was still taught, it wasn’t until I got to university I discovered that actually there are many intergrands that cannot be integrated and that the trapezium rule is an excellent method for approximation. This was a fact I’d forgotten between university and entering the teaching profession, but a fact Tom reminded me of.

This seems to me to be a very good reason to keep the trapezium rule in the syllabus. I was teaching it last week and I was thinking about this, and I realised that the way we assess the trapezium rule at A level is silly. We always ask students to approximate an integral than integrate it using calculus, oven via substitution or parts. This can only add to the feeling among students that the trapezium rule is pointless, as they can instantly see a way to find a much more accurate value. I now make a concerted effort to examine it’s importance and to state why I feel it gets a bad run, this had a positive effect on my class this year and they were much more engaged with it than previous classes.

This is not the only numerical method that gets a poor deal on our exams, another that jumps to mind is trial and improvement, a simple iteration method that can be used to find a reasonably accurate solution to an equation, however at GCSE the equation is often a quadratic, which students can find an actual solution to relatively easily via the formula or by completing the square. Why not use an equation they can’t solve otherwise?!

What are your views on numerical methods? Have you had similar thoughts? Is there anything you used to dislike teaching but have changed your mind on? If so, I’d love to hear in the comments.

I understand what you mean – perhaps it’s just some additional wording which would help – I always emphasise that by working out the exact answer we can compare the numerical methods with answers by exact methods to show how good the numerical methods are. So we can have confidence in the numerical methods when we can’t work out the exact answer. Does that make sense?

That does make sense, however how you do the comparison is important. I could create a method that integrates cubics incredibly well (say exactly) but exhibits very odd behaviour with arbitrary functions.

Leo

February 18, 2016 at 11:20 pm

When you compare the trapezium rule approximations to a known integral and increase the number of strips, second order convergence on the known integral is seen which, when working with integrands you can’t integrate analytically, allows you to extrapolate with more iterations or theoretically (using the sum of a geometric progression). This gets you to some pretty accurate approximations.

It certainly does, we can get amazingly accurate approximations. And this is what we need to be sharing. I fear that sometimes it’s framed as it was to me way back when, that these methods have no worth. They do, they really do.

It all starts in primary school when they get you to draw round your hand on squared paper and count all the full squares first, then shade in all the partial squares that are more than half within your handprint and discard the others because, you know, they cancel each other out. Do they? Do they really? Did anyone in year 3 count all the partial ones discarded and counted and get the same number? Clearly not, and so students get the notion to follow a few rules from the teacher and believe what they say without question.

I like to introduce the trapezium rule in C2 by asking students to work out the area ‘under the graph’ of 1/x between two limits. Dutifully, they attempt to integrate, but then realise they cannot, as it leads to a division by zero – a nice non-example of the type of integral previously studied. Nevertheless, I still want to know the area, so an alternative method must be sought. Of course, it is possible to integrate 1/x, but not using the methods encountered up to that point of Y12 (with Edexcel at least) so mentioning this serves as a taster for the fact that more involved methods will be required to perform integration (or differentiation) for more complex functions, which is a good chunk of A2 maths.

Numerical methods and more broadly “applied mathematics”, for whatever reason, have been a polarizing subject within the mathematics community. Some “purists” poo-poo it and similarly, some applied mathematicians view “pure math” as an excessively abstract construct. Interesting is theoretical applied mathematics, wherein there is a theoretical study of numerical methods. Numerical methods for partial differential equations is rich in this area of theoretical study (among others).